A Beurling-Chen-Hadwin-Shen Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras with Unitarily Invariant Norms

TL;DR

This paper extends the Beurling-Chen-Hadwin-Shen theorem to noncommutative Hardy spaces associated with semifinite von Neumann algebras, characterizing invariant subspaces under a broad class of unitarily invariant norms.

Contribution

It introduces a new class of unitarily invariant norms and proves a generalized Beurling-type theorem for noncommutative Hardy spaces in this setting.

Findings

Characterization of $H^e$-invariant subspaces in noncommutative $L^e$ spaces.

Extension of the theorem to crossed product von Neumann algebras.

Application to invariant subspaces in noncommutative Banach function spaces.

Abstract

We introduce a class of unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norms with repect to $\tau$ on a von Neumann algebra $\mathcal{M}$ with a faithful, normal, semifinite tracial weight $\tau$. We prove a Beurling-Chen-Hadwin-Shen theorem for $H^\infty$-invariant spaces of $L^\alpha(\mathcal{M},\tau)$, where $\alpha$ is a unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuous norm with respect to $\tau$, and $H^\infty$ is an extension of Arveson's noncommutative Hardy space. We use our main result to characterize the $H^\infty$-invariant subspaces of a noncommutative Banach function space $\mathcal I(\tau)$ with the norm $\|\cdot\|_{E }$ on $\mathcal{M}$, the crossed product of a semifinite von Neumann algebra by an action $\beta$, and $B(\mathcal{H})$ for a separable Hilbert space $\mathcal{H}$.